Developing proof skills So I’m a freshman taking an intro to analysis class and I love it but the thought process behind developing arguments to prove concepts and theorems is difficult.
I understand the basics of the  logical framework I have to follow but I find that I tend to get a lightbulb moment that is followed by vigorous writing but then the idea kind of disappears and this will happen several times while working through a problem. 
Today I tried something new where I had my friend act as my pupil and I would show them my thought process with diagrams and analogies. It was very fun and my ideas felt more fluid. I understand that as I take more proof based classes (this is my first) that I will become more accustomed to the thinking style but for now, how can I simulate the teaching by learning method without an actual person sitting in front of me?
 A: In his book "Hilbert Space Problem Book", P.R.Halmos wrote in the introduction: "The only way to learn mathematics is to do mathematics."  You cannot learn to write proofs by only reading someone else's proofs; you cannot learn to write proofs by only listening to someone who knows how to write proofs; you can learn to write proofs only by writing them yourselves. Granted, they will look awkward at start, be sprinkled with little and big mistakes, but they will become better and better as you go. Every correct proof you write from scratch is owned by you; it is wired into your system and it will take a long time to forget it. On the other hand, a proof you read through will never be owned by you; it will never get wired into your system, and it will take only a few days to forget it.
I found that the best way to practice mathematics is to close all sources and do it myself. I didn't succeed every time, but I am pretty sure that the difference between people who really understand mathematics and all the rest lies exactly in the amount of (correct) proofs they were able to write down themselves.
